Hybrid scheme for Brownian semistationary processes

Transcription

1 Hybrid scheme for Browia semistatioary processes Mikkel Beedse Asger Lude Mikko S. Pakkae September 8, 28 arxiv:57.34v4 [math.pr] 4 May 27 Abstract We itroduce a simulatio scheme for Browia semistatioary processes, which is based o discretizig the stochastic itegral represetatio of the process i the time domai. We assume that the kerel fuctio of the process is regularly varyig at zero. The ovel feature of the scheme is to approximate the kerel fuctio by a power fuctio ear zero ad by a step fuctio elsewhere. The resultig approximatio of the process is a combiatio of Wieer itegrals of the power fuctio ad a Riema sum, which is why we call this method a hybrid scheme. Our mai theoretical result describes the asymptotics of the mea square error of the hybrid scheme ad we observe that the scheme leads to a substatial improvemet of accuracy compared to the ordiary forward Riema-sum scheme, while havig the same computatioal complexity. We exemplify the use of the hybrid scheme by two umerical experimets, where we examie the fiite-sample properties of a estimator of the roughess parameter of a Browia semistatioary process ad study Mote Carlo optio pricig i the rough Bergomi model of Bayer et al. [], respectively. Keywords: Stochastic simulatio; discretizatio; Browia semistatioary process; stochastic volatility; regular variatio; estimatio; optio pricig; rough volatility; volatility smile. JEL Classificatio: C22, G3, C3 MSC 2 Classificatio: 6G2, 6G22, 65C2, 9G6, 62M9 Itroductio We study simulatio methods for Browia semistatioary BSS processes, first itroduced by Bardorff-Nielse ad Schmiegel [8, 9], which form a flexible class of stochastic processes that are able to capture some commo features of empirical time series, such as stochastic volatility itermittecy, roughess, statioarity ad strog depedece. By ow these processes have bee Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Mathematics, Imperial College Lodo, South Kesigto Campus, Lodo SW7 2AZ, UK ad CREATES, Aarhus Uiversity, Demark.

2 applied i various cotexts, most otably i the study of turbulece i physics [7, 6] ad i fiace as models of eergy prices [4, ]. A BSS process X is defied via the itegral represetatio Xt = t gt sσsdw s,. where W is a two-sided Browia motio providig the fudametal oise iovatios, the amplitude of which is modulated by a stochastic volatility itermittecy process σ that may deped o W. This drivig oise is the covolved with a determiistic kerel fuctio g that specifies the depedece structure of X. The process X ca also be viewed as a movig average of volatilitymodulated Browia oise ad settig σs =, we see that statioary Browia movig averages are ested i this class of processes. I the applicatios metioed above, the case where X is ot a semimartigale is particularly relevat. This situatio arises whe the kerel fuctio g behaves like a power-law ear zero; more specifically, whe for some α 2, 2 \ {}, gx x α for small x >..2 Here we write to idicate proportioality i a iformal sese, aticipatig a rigorous formulatio of this relatioship give i Sectio 2.2 usig the theory of regular variatio [5], which plays a sigificat role i our subsequet argumets. The case α = 6 i.2 is importat i statistical modelig of turbulece [6] as it gives rise to processes that are compatible with Kolmogorov s scalig law for ideal turbulece. Moreover, processes of similar type with α.4 have bee recetly used i the cotext of optio pricig as models of rough volatility [,, 8, 2], see Sectios 2.5 ad 3.3 below. The case α = would roughly speakig lead to a process that is a semimartigale, which is thus excluded. Uder.2, the trajectories of X behave locally like the trajectories of a fractioal Browia motio with Hurst idex H = α + 2, \ { 2 }. While the local behavior ad roughess, measured i terms of Hölder regularity, of X are determied by the parameter α, the global behavior of X e.g., whether the process has log or short memory depeds o the behavior of gx as x, which ca be specified idepedetly of α. This should be cotrasted with fractioal Browia motio ad related self-similar models, which ecessarily must coform to a restrictive affie relatioship betwee their Hölder regularity local behavior ad roughess ad Hurst idex global behavior, as elucidated by Geitig ad Schlather [2]. Ideed, i the realm of BSS processes, local ad global behavior are coveietly decoupled, which uderlies the flexibility of these processes as a modelig framework. I coectio with practical applicatios, it is importat to be able to simulate the process X. If the volatility process σ is determiistic ad costat i time, the X will be strictly statioary ad Gaussia. This makes X ameable to exact simulatio usig the Cholesky factorizatio or circulat embeddigs, see, e.g., [2, Chapter XI]. However, it seems difficult, if ot impossible, to develop a exact method that is applicable with a stochastic σ, as the process X is the either Markovia or Gaussia. Thus, i the geeral case oe must resort to approximative methods. To this ed, Beth et al. [3] have recetly proposed a Fourier-based method of simulatig BSS 2

3 processes, ad more geeral Lévy semistatioary LSS processes, which relies o approximatig the kerel fuctio g i the frequecy domai. I this paper, we itroduce a ew discretizatio scheme for BSS processes based o approximatig the kerel fuctio g i the time domai. Our startig poit is the Riema-sum discretizatio of.. The Riema-sum scheme builds o a approximatio of g usig step fuctios, which has the pitfall of failig to capture appropriately the steepess of g ear zero. I particular, this becomes a serious defect uder.2 whe α 2,. I our ew scheme, we mitigate this problem by approximatig g usig a appropriate power fuctio ear zero ad a step fuctio elsewhere. The resultig discretizatio scheme ca be realized as a liear combiatio of Wieer itegrals with respect to the drivig Browia motio W ad a Riema sum, which is why we call it a hybrid scheme. The hybrid scheme is oly slightly more demadig to implemet tha the Riema-sum scheme ad the schemes have the same computatioal complexity as the umber of discretizatio cells teds to ifiity. Our mai theoretical result describes the exact asymptotic behavior of the mea square error MSE of the hybrid scheme ad, as a special case, that of the Riema-sum scheme. We observe that switchig from the Riema-sum scheme to the hybrid scheme reduces the asymptotic root mea square error RMSE substatially. Usig merely the simplest variat the of hybrid scheme, where a power fuctio is used i a sigle discretizatio cell, the reductio is at least 5% for all α, 2 ad at least 8% for all α 2,. The reductio i RMSE is close to % as α approches 2, which idicates that the hybrid scheme ideed resolves the problem of poor precisio that affects the Riema-sum scheme. To assess the accuracy of the hybrid scheme i practice, we perform two umerical experimets. Firstly, we examie the fiite-sample performace of a estimator of the roughess idex α, itroduced by Bardorff-Nielse et al. [6] ad Corcuera et al. [6]. This experimet eables us to assess how faithfully the hybrid scheme approximates the fie properties of the BSS process X. Secodly, we study Mote Carlo optio pricig i the rough Bergomi stochastic volatility model of Bayer et al. []. We use the hybrid scheme to simulate the volatility process i this model ad we fid that the resultig implied volatility smiles are idistiguishable from those simulated usig a method that ivolves exact simulatio of the volatility process. Thus we are able propose a solutio to the problem of fidig a efficiet simulatio scheme for the rough Bergomi model, left ope i the paper []. The rest of this paper is orgaized as follows. I Sectio 2 we recall the rigorous defiitio of a BSS process ad itroduce our assumptios. We also itroduce the hybrid scheme, state our mai theoretical result cocerig the asymptotics of the mea square error ad discuss a extesio of the scheme to a class of trucated BSS processes. Sectio 3 briefly discusses the implemetatio of the discretizatio scheme ad presets the umerical experimets metioed above. Fially, Sectio 4 cotais the proofs of the theoretical ad techical results give i the paper. 3

16 usig the radom vectors {Wi } T i= ad radom variables {σi } T i=. I the hybrid scheme, it typically suffices to take κ to be at most 3. Thus, i 3.4, the first sum ˇX i requires oly a egligible computatioal effort. By cotrast, the umber of terms i the secod sum ˆX i icreases as. It is the useful to ote that i N ˆX = Γ k Ξ i k = Γ Ξ i, k= where, k =,..., κ, Γ k := g b k, k = κ +, κ + 2,..., N, Ξ k := σ k W k, k = N, N +,..., T. ad Γ Ξ stads for the discrete covolutio of the sequeces Γ ad Ξ. It is well-kow that the discrete covolutio ca be evaluated efficietly usig a fast Fourier trasform FFT. The computatioal complexity of simultaeously evaluatig Γ Ξ i for all i =,,..., T usig a FFT is ON log N, see [23, pp. 79 8], which uder A4 traslates to O γ+ log. The computatioal complexity of the etire hybrid scheme is the O γ+ log, provided that {σi } T i= N is geerated usig a scheme with complexity ot exceedig O γ+ log. As a compariso, we metio that the complexity of a exact simulatio of a statioary Gaussia process usig circulat embeddigs is O log [2, p. 36], whereas the complexity of the Cholesky factorizatio is O 3 [2, p. 32]. Remark 3.2. With T BSS processes, the computatioal complexity of the hybrid scheme via 3.7 is O log. Figure 2 presets examples of trajectories of the BSS process X usig the hybrid scheme with κ =, 2 ad b = b. We choose the kerel fuctio g to be the gamma kerel Example 2.3 with λ =. We also discretize X usig the Riema-sum scheme, κ = with b {b FWD, b } that is, the forward Riema-sum scheme ad its couterpart with optimally chose evaluatio poits. We ca make two observatios: Firstly, we see how the roughess parameter α cotrols the regularity properties of the trajectories of X as we decrease α, the trajectories of X become icreasigly rough. Secodly, ad more importatly, we see how the simulated trajectories comig from the Riema-sum ad hybrid schemes ca be rather differet, eve though we use the same iovatios for the drivig Browia motio. I fact, the two variats of the hybrid scheme κ =, 2 yield almost idetical trajectories, while the Riema-sum scheme κ = produces trajectories that are comparatively smoother, this differece becomig more apparet as α approaches 2. Ideed, i the extreme case with α =.499, both variats of the Riema-sum scheme break dow ad yield aomalous trajectories with very little variatio, while the hybrid scheme cotiues to produce accurate results. The fact that the hybrid scheme is able to reproduce the fie properties of rough BSS processes, eve for values of α very close to 2, is backed up by a further experimet reported i the followig sectio. 6

18 3.2 Estimatio of the roughess parameter Suppose that we have observatios X i m, i =,,..., m, of the BSS process X, give by 2., for some m N. Bardorff-Nielse et al. [6] ad Corcuera et al. [6] discuss how the roughess idex α ca be estimated cosistetly as m. The method is based o the chage-of-frequecy COF statistics COFX, m = m k=5 m k=3 X k m 2X k 2 m + X k 4 2 m X k m 2X k m + X k 2 2, m 5, m which compare the realized quadratic variatios of X, usig secod-order icremets, with two differet lag legths. Corcuera et al. [6] have show that uder some assumptios o the process X, which are similar to A, A2 ad A3 albeit slightly more restrictive, it holds that ˆαX, m := log COFX, m P α, m log 2 2 A i-depth study of the fiite sample performace of this COF estimator ca be foud i [2]. To examie how well the hybrid scheme reproduces the fie properties of the BSS process i terms of regularity/roughess, we apply the COF estimator to discretized trajectories of X, where the kerel fuctio g is agai the gamma kerel Example 2.3 with λ =, geerated usig the hybrid scheme with κ =, 2, 3 ad b = b. We cosider the case where the volatility process satisfies σt =, that is, the process X is Gaussia. This allows us to quatify ad cotrol for the itrisic bias ad oisiess, measured i terms of stadard deviatio, of the estimatio method itself, by iitially applyig the estimator to trajectories that have bee simulated usig a exact method based o the Cholesky factorizatio. We the study the behavior of the estimator whe applied to a discretized trajectory, while decreasig the step size of the discretizatio scheme. More precisely, we simulate ˆαX, m, where m = 5 ad X is the hybrid scheme for X with = ms ad s {, 2, 5}. This meas that we compute ˆαX, m usig m observatios obtaied by subsamplig every s-th observatio i the sequece X i, i =,,...,. As a compariso, we repeat these simulatios substitutig the hybrid scheme with the Riema-sum scheme, usig κ = with b {b FWD, b }. The results are preseted i Figure 3. We observe that the itrisic bias of the estimator with m = 5 observatios is egligible ad hece the bias of the estimates computed from discretized trajectories is the attributable to approximatio error arisig from the respective discretizatio scheme, where positive resp. egative bias idicates that the simulated trajectories are smoother resp. rougher tha those of the process X. Cocetratig first o the baselie case s =, we ote that the hybrid scheme produces essetially ubiased results whe α 2,, while there is moderate bias whe α, 2, which disappears whe passig from κ = to κ = 3, eve for values of α very close to 2. The largest value of α cosidered i our simulatios is α =.49; oe would expect the performace to weake as α approaches 2, cf. Figure, but this rage of parameter values seems to be of limited practical iterest. The stadard deviatios exhibit a similar patter. The correspodig results for the Riema-sum scheme are clearly iferior, exhibitig sigificat bias, while usig optimal evaluatio poits b = b improves the situatio slightly. I particular, 8

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